Approximability of Capacitated Network Design

In the capacitated survivable network design problem (Cap-SNDP), we are given an undirected multi-graph where each edge has a capacity and a cost. The goal is to find a minimum cost subset of edges that satisfies a given set of pairwise minimum-cut requirements. Unlike its classical special case of SNDP when all capacities are unit, the approximability of Cap-SNDP is not well understood; even in very restricted settings no known algorithm achieves a o ( m ) approximation, where m is the number of edges in the graph. In this paper, we obtain several new results and insights into the approximability of Cap-SNDP. We give an O (log n ) approximation for a special case of Cap-SNDP where the global minimum cut is required to be at least R . (Note that this problem generalizes the min-cost λ -edge-connected subgraph problem, which is the special case of our problem when all capacities are unit.) Our result is based on a rounding of a natural cut-based LP relaxation strengthened with knapsack-cover (KC) inequalities. Our technique extends to give a similar approximation for a new network design problem that captures global minimum cut as a special case. We then show that as we move away from global connectivity, even for the single pair case (that is, when only one pair ( s , t ) has positive connectivity requirement), this strengthened LP has Ω ( n ) integrality gap. We also consider a variant of Cap-SNDP in which multiple copies of an edge can be bought: we give an O (log k ) approximation for this case, where k is the number of vertex pairs with non-zero connectivity requirement. This improves upon the previously known O (min{ k ,log R max })-approximation when R max is large; here R max is the largest requirement. On the other hand, we observe that the multiple copy version of Cap-SNDP is Ω (loglog n )-hard to approximate even for the single-source version of the problem.